"Everyone wants to restore relations in the Western community after - thanks to the former U.S. administration - many European countries adopted an anti-Trump position during the election campaign", he said. Right now we're not getting along with Russian Federation at all.
Mr Trump has voiced caution on his relationship with Mr Putin, and said he would wait and see about future ties with Moscow.
"This has built for a long period of time".
"We know that we all need to contribute our fair share", Stoltenberg said. Tillerson is a former oil executive who was awarded Russia's Order of Friendship by Putin.
Ten countries voted in favour of the text, while Bolivia joined Russian Federation in voting no.
He said the visit - the first to Moscow by a senior Trump administration official - offered an opportunity to clarify the chances of cooperation "above all on the formation of a broad anti-terrorist front".
"In general, primitiveness and loutishness are very characteristic of the current rhetoric coming out of Washington", Deputy Foreign Minister Sergei Ryabkov told Russia's state-owned RIA news agency. "The obstacles in the way of a reset now are more serious than you had at the outset of any other administration since the end of the Cold War".
"I don't remember a case of a dictator being removed smoothly without violence", said Lavrov, lamenting past efforts to remove Slobodan Milosevic from Serbia, Saddam Hussein from Iraq, and Moammar Gadhafi from Libya.
Trump hoped that the European countries would have nothing to fear from Russian Federation. "I would like to think that they didn't know, but certainly they could have". Right now, there is a fear.
Tillerson noted the low level of trust between the two countries.
"We believe that the precondition for a political dialogue with Russian Federation is that we are strong and that we are united". That based on that, we can talk to Russian Federation because Russia is our neighbour. Russian Federation is here to stay. "We're going to see how that all works out", Trump said.
"I liked how we spent yesterday".
The U.S. has blamed the Syrian government for launching the attack that killed more than 80, while Russian Federation has claimed that toxic agents were released from a rebel chemical arsenal hit by Syrian warplanes.
Asked whether Assad could be subject to war crimes charges, Tillerson said people were working to make such a case.
During Tillerson's visit to Moscow on Wednesday, Russian Foreign Minister Sergei Lavrov gave the top USA diplomat an earful of criticism over the US strike.
Nikki Haley, the US Ambassador to the United Nations, had expressed the same sentiment before the chemical attack. And that means that we have to invest in our collective defence.
It was the eighth time during Syria's civil war that Moscow has used its veto power on the Security Council to shield Assad's government. Because Russia will not go away.
At the start of his meeting with Lavrov, Tillerson said he wanted to "clarify areas of common objectives, areas of common interest - even where our tactical approaches may be different - and further clarify areas of sharp difference".
This is published unedited from the PTI feed.